Reduced complexity sliding window based equalizer

ABSTRACT

A method and apparatus for use in data estimation in wireless communication are provided. A wireless communications signal is received and transformed to produce a received vector. The received vector is processed using a sliding window based approach that includes processing each of a plurality of windows. For each window, an approximate circulant channel response matrix is produced for use in estimating a data vector corresponding to the window.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 12/191,702 filed on Aug. 14, 2008, which was a continuation of U.S. patent application Ser. No. 11/265,932, filed on Nov. 3, 2005, which was a continuation of U.S. patent application Ser. No. 10/875,900, filed on Jun. 24, 2004, now U.S. Pat. No. 7,042,967, which claims priority from U.S. Provisional Application No. 60/482,333, filed on Jun. 25, 2003. U.S. patent Ser. No. 10/875,900 was also a continuation-in-part of U.S. patent application Ser. No. 10/791,244, filed Mar. 2, 2004, which claims priority from 60/452,165, filed on Mar. 3, 2003. All of the above applications are incorporated by reference as if fully set forth.

FIELD OF INVENTION

The invention generally relates to wireless communication systems, In particular, the invention relates to data detection in such systems.

BACKGROUND

Due to the increased demands for improved receiver performance, many advanced receivers use zero forcing (ZF) block linear equalizers and minimum mean square error (MMSE) equalizers.

In both these approaches, the received signal is typically modeled per Equation 1.

r=Hd+n  Equation 1

r is the received vector, comprising samples of the received signal. H is the channel response matrix. d is the data vector to be estimated. In spread spectrum systems, such as code division multiple access (CDMA) systems, d may be represented as data symbols or a composite spread data vector. For a composite spread data vector, the data symbols for each individual code are produced by despreading the estimated data vector d with that code. n is the noise vector.

In a ZF block linear equalizer, the data vector is estimated, such as per Equation 2.

d=(H ^(H) H)⁻¹ H ^(H) r  Equation 2

(·)^(H) is the complex conjugate transpose (or Hermitian) operation. In a MMSE block linear equalizer, the data vector is estimated, such as per Equation 3.

d=(H ^(H) H+σ ² I)⁻¹ H ^(H) r  Equation 3

In wireless channels experiencing multipath propagation, to accurately detect the data using these approaches requires that an infinite number of received samples be used, which is not practical. Therefore, it is desirable to use an approximation technique. One of the approaches is a sliding window approach. In the sliding window approach, a predetermined window of received samples and channel responses are used in the data detection. After the initial detection, the window is slid down to a next window of samples. This process continues until the communication ceases.

By not using an infinite number of samples, an error is introduced into the symbol model shown in Equation 1 and, therefore causes inaccurate data detection. The error is most prominent at the beginning and end of the window, where the effectively truncated portions of the infinite sequence have the largest impact. One approach to reduce these errors is to use a large window size and truncate the results at the beginning and the end of the window. The truncated portions of the window are determined in previous and subsequent windows. This approach has considerable complexity, especially when the channel delay spread is large. The large window size leads to large dimensions on the matrices and vectors used in the data estimation. Additionally, this approach is not computationally efficient by detection data at the beginning and at the ends of the window and then discarding that data.

Accordingly, it is desirable to have alternate approaches to data detection.

SUMMARY

A method and apparatus for use in data estimation in wireless communication are provided. A wireless communications signal is received and transformed to produce a received vector. The received vector is processed using a sliding window based approach that includes processing each of a plurality of windows. For each window, an approximate circulant channel response matrix is produced for use in estimating a data vector corresponding to the window.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of a banded channel response matrix.

FIG. 2 is an illustration of a center portion of the banded channel response matrix.

FIG. 3 is an illustration of a data vector window with one possible partitioning.

FIG. 4 is an illustration of a partitioned signal model.

FIG. 5 is a flow diagram of sliding window data detection using a past correction factor.

FIG. 6 is a receiver using sliding window data detection using a past correction factor.

FIG. 7 is a flow diagram of sliding window data detection using a noise auto-correlation correction factor.

FIG. 8 is a receiver using sliding window data detection using a noise auto-correlation correction factor.

FIG. 9 is a graphical representation of the sliding window process.

FIG. 10 is a graphical representation of the sliding window process using a circulant approximation.

FIG. 11 is a circuit for an embodiment for detecting data using discrete Fourier transforms (DFTs).

DETAILED DESCRIPTION

Although the features and elements of the present invention are described in the preferred embodiments in particular combinations, each feature or element can be used alone (without the other features and elements of the preferred embodiments) or in various combinations with or without other features and elements of the present invention.

Hereafter, a wireless transmit/receive unit (WTRU) includes but is not limited to a user equipment, mobile station, fixed or mobile subscriber unit, pager, or any other type of device capable of operating in a wireless environment. When referred to hereafter, a base station includes but is not limited to a Node-B, site controller, access point or any other type of interfacing device in a wireless environment.

Although reduced complexity sliding window equalizer is described in conjunction with a preferred wireless code division multiple access communication system, such as CDMA2000 and universal mobile terrestrial system (UMTS) frequency division duplex (FDD), time division duplex (TDD) modes and time division synchronous CDMA (TD-SCDMA), it can be applied to various communication system and, in particular, various wireless communication systems. In a wireless communication system, it can be applied to transmissions received by a WTRU from a base station, received by a base station from one or multiple WTRUs or received by one WTRU from another WTRU, such as in an ad hoc mode of operation.

The following describes the implementation of a reduced complexity sliding window based equalizer using a preferred MMSE algorithm. However, other algorithms can be used, such as a zero forcing algorithm. h(·) is the impulse response of a channel. d(k) is the k^(th) transmitted sample that is generated by spreading a symbol using a spreading code. It can also be sum of the chips that are generated by spreading a set of symbols using a set of codes, such as orthogonal codes. r(·) is the received signal. The model of the system can expressed as per Equation 4.

$\begin{matrix} {{r(t)} = {{{\sum\limits_{k = {- \infty}}^{\infty}\; {{d(k)}{h\left( {t - {kT}_{c}} \right)}}} + {n(t)}\mspace{31mu} - \infty} < t < \infty}} & {{Equation}\mspace{20mu} 4} \end{matrix}$

n(t) is the sum of additive noise and interference (intra-cell and inter-cell). For simplicity, the following is described assuming chip rate sampling is used at the receiver, although other sampling rates may be used, such as a multiple of the chip rate. The sampled received signal can be expressed as per Equation 5.

$\begin{matrix} {\mspace{20mu} {{{r(j)} = {{{\sum\limits_{k = {- \infty}}^{\infty}\; {{d(k)}{h\left( {j - k} \right)}}} + {n(j)}} = {{\sum\limits_{k = {- \infty}}^{\infty}\; {{d\left( {j - k} \right)}{h(k)}}} + {n(j)}}}},\mspace{79mu} {j \in \left\{ {\ldots \;,{- 2},{- 1},0,1,2,\ldots}\; \right\}}}} & {{Equation}\mspace{20mu} 5} \end{matrix}$

T_(c) is being dropped for simplicity in the notations.

Assuming h(·) has a finite support and is time invariant. This means that in the discrete-time domain, index L exists such that h(i)=0 for i<0 and i≧L. As a result, Equation 5 can be re-written as Equation 6.

$\begin{matrix} {{{r(j)} = {{\sum\limits_{k = 0}^{L - 1}\; {{h(k)}{d\left( {j - k} \right)}}} + {n(j)}}},{j \in \left\{ {\ldots \;,{- 2},{- 1},0,1,2,\ldots}\; \right\}}} & {{Equation}\mspace{20mu} 6} \end{matrix}$

Considering that the received signal has M received signals r(0), . . . , r(M−1), Equation 7 results.

r=Hd+n

where,

r=[r(0), . . . , r(M−1)]^(T)εC^(M),

d=[d(−L+1), d(−L+2), . . . , d(0), d(1), . . . , d(M−1)]^(T)εC^(M+L−1)

n=[n(0), . . . , n(M−1)]^(T)εC^(M)

$\begin{matrix} \; & {\mspace{641mu} {{Equation}\mspace{14mu} 7}} \end{matrix}$ $H = {\left\lbrack \begin{matrix} {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(1)} & {h(0)} & 0 & \cdots & \cdots \\ 0 & {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(1)} & {h(0)} & 0 & \cdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \cdots & \cdots & 0 & {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(1)} & {h(0)} \end{matrix} \right\rbrack  \in C^{M \times {({M + L - 1})}}}$

In Equation 7, C^(M) represents the space of all complex vectors with dimension M.

Part of the vector d can be determined using an approximate equation. Assuming M>L and defining N=M−L+1, vector d is per Equation 8.

$\begin{matrix} {d = {\begin{bmatrix} \underset{L - 1}{\underset{}{{d\left( {{- L} + 1} \right)},{d\left( {{- L} + 2} \right)},\ldots \mspace{14mu},{d\left( {- 1} \right)}},} \\ \underset{N}{\underset{}{{d(0)},{d(1)},\ldots \mspace{11mu},{d\left( {N - 1} \right)}},} \\ \underset{\underset{L - 1}{}}{{d(N)},\ldots \mspace{11mu},{d\left( {N + L - 2} \right)}} \end{bmatrix}^{T} \in C^{N + {2L} - 2}}} & {{Equation}\mspace{20mu} 8} \end{matrix}$

The H matrix in Equation 7 is a banded matrix, which can be represented as the diagram in FIG. 1. In FIG. 1, each row in the shaded area represents the vector [h(L−1), h(L−2), . . . , h(1), h(0)], as shown in Equation 7.

Instead of estimating all of the elements in d, only the middle N elements of d are estimated. {tilde over (d)} is the middle N elements as per Equation 9.

{tilde over (d)}=[d(0), . . . , d(N−1)]^(T)  Equation 9

Using the same observation for r, an approximate linear relation between r and {tilde over (d)} is per Equation 10.

r={tilde over (H)}{tilde over (d)}+n Equation 10

Matrix {tilde over (H)} can be represented as the diagram in FIG. 2 or as per Equation 11.

$\begin{matrix} {\overset{\sim}{H} = \begin{bmatrix} {h(0)} & 0 & \cdots & \; \\ {h(1)} & {h(0)} & \ddots & \; \\ \vdots & {h(1)} & \ddots & 0 \\ {h\left( {L - 1} \right)} & \vdots & \ddots & {h(0)} \\ 0 & {h\left( {L - 1} \right)} & \ddots & {h(1)} \\ \vdots & 0 & \ddots & \vdots \\ \; & \vdots & \ddots & {h\left( {L - 1} \right)} \end{bmatrix}} & {{Equation}\mspace{20mu} 11} \end{matrix}$

As shown, the first L−1 and the last L−1 elements of r are not equal to the right hand side of the Equation 10. As a result, the elements at the two ends of vector {tilde over (d)} will be estimated less accurately than those near the center. Due to this property, a sliding window approach, as described subsequently, is preferably used for estimation of transmitted samples, such as chips.

In each k^(th) step of the sliding window approach, a certain number of the received samples are kept in r [k] with dimension N+L−1. They are used to estimate a set of transmitted data {tilde over (d)}[k] with dimension N using equation 10. After vector {tilde over (d)}[k] is estimated, only the “middle” part of the estimated vector {tilde over ({circumflex over (d)}[k] is used for the further data processing, such as by despreading. The “lower” part (or the later in-time part) of {tilde over (d)}[k] is estimated again in the next step of the sliding window process in which r [k+1] has some of the elements r [k] and some new received samples, i.e. it is a shift (slide) version of r [k].

Although, preferably, the window size N and the sliding step size are design parameters, (based on delay spread of the channel (L), the accuracy requirement for the data estimation and the complexity limitation for implementation), the following using the window size of Equation 12 for illustrative purposes.

N=4N _(s)×SF  Equation 12

SF is the spreading factor. Typical window sizes are 5 to 20 times larger than the channel impulse response, although other sizes may be used.

The sliding step size based on the window size of Equation 12 is, preferably, 2N_(s)×SF. N_(s)ε{1, 2, . . . } is, preferably, left as a design parameter. In addition, in each sliding step, the estimated chips that are sent to the despreader are 2N_(s)×SF elements in the middle of the estimated {circumflex over (d)}[k]. This procedure is illustrated in FIG. 3.

In the sliding window approach described above, the system model is approximated by throwing away some terms in the model. In the following, a technique is described where terms are kept by either using the information estimated in previous sliding step or characterizing the terms as noise in the model. The system model is corrected using the kept/characterized terms.

One algorithm of data detection uses an MMSE algorithm with model error correction uses a sliding window based approach and the system model of Equation 10.

Due to the approximation, the estimation of the data, such as chips, has error, especially, at the two ends of the data vector in each sliding step (the beginning and end). To correct this error, the H matrix in Equation 7 is partitioned into a block row matrix, as per Equation 13, (step 50).

H=[H _(p) |{tilde over (H)}|H _(f)]  Equation 13

Subscript “p” stands for “past”, and “f” stands for “future”. {tilde over (H)} is as per Equation 10. H_(p) is per Equation 14.

$\begin{matrix} {H_{p} = {\begin{bmatrix} {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(1)} \\ 0 & {h\left( {L - 1} \right)} & \cdots & {h(2)} \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & {h\left( {L - 1} \right)} \\ 0 & \cdots & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & \cdots & \cdots & 0 \end{bmatrix} \in C^{{({N + L - 1})} \times {({L - 1})}}}} & {{Equation}\mspace{20mu} 14} \end{matrix}$

H_(f) is per Equation 15.

$\begin{matrix} {H_{f} = {\begin{bmatrix} 0 & \cdots & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & \cdots & \cdots & 0 \\ {h(0)} & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ {h\left( {L - 3} \right)} & \cdots & {h(0)} & 0 \\ {h\left( {L - 2} \right)} & {h\left( {L - 3} \right)} & \cdots & {h(0)} \end{bmatrix} \in C^{{({N + L - 1})} \times {({L - 1})}}}} & {{Equation}\mspace{20mu} 15} \end{matrix}$

The vector d is also partitioned into blocks as per Equation 16.

d=[d _(p) ^(T) |{tilde over (d)} ^(T) |d _(f) ^(T)]^(T)  Equation 16

{tilde over (d)} is the same as per Equation 8 and d_(p) is per Equation 17.

d _(p) =[d(−L+1)d(−L+2) . . . d(−1)]^(T) εC ^(L-1)  Equation 17

d_(f) is per Equation 18.

d _(f) =[d(N)d(N+1) . . . d(N+L−2)]^(T) εC ^(L-1)  Equation 18

The original system model is then per Equation 19 and is illustrated in FIG. 4.

r=H _(p) d _(p) +{tilde over (H)}{tilde over (d)}+H _(f) d _(f) +n  Equation 19

One approach to model Equation 19 is per Equation 20.

{tilde over (r)}={tilde over (H)}{tilde over (d)}+ñ ₁  Equation 20

where {tilde over (r)}=r−H_(p)d_(p) and ñ₁=H_(f)d_(f)+n

Using an MMSE algorithm, the estimated data vector {tilde over ({circumflex over (d)} is per Equation 21.

{tilde over ({circumflex over (d)}=g _(d) {tilde over (H)} ^(H)(g _(d) {tilde over (H)}{tilde over (H)} ^(H)+Σ₁)⁻¹ {tilde over ({circumflex over (r)}  Equation 21

In Equation 21, g_(d) is chip energy per Equation 22.

E{d(i)d*(j)}=g _(d)δ_(ij)  Equation 22

{tilde over ({circumflex over (r)} is per Equation 23.

{tilde over ({circumflex over (r)}=r−H _(p) {circumflex over (d)} _(p)  Equation 23

{circumflex over (d)}_(p), is part of the estimation of {tilde over (d)} in the previous sliding window step. Σ₁ is the autocorrelation matrix of ñ₁, i.e., Σ₁=E{ñ₁ñ₁ ^(H)}. If assuming H_(f)d_(f) and n are uncorrelated, Equation 24 results.

Σ₁ =g _(d) H _(f) H _(f) ^(H) +E{nn ^(H)}  Equation 24

The reliability of depends on the sliding window size (relative to the channel delay span L) and sliding step size.

This approach is also described in conjunction with the flow diagram of FIG. 5 and preferred receiver components of FIG. 6, which can be implemented in a WTRU or base station. The circuit of FIG. 6 can be implemented on a single integrated circuit (IC), such as an application specific integrated circuit (ASIC), on multiple IC's, as discrete components or as a combination of IC('s) and discrete components.

A channel estimation device 20 processes the received vector r producing the channel estimate matrix portions, H_(p), {tilde over (H)} and H_(f), (step 50). A future noise auto-correlation device 24 determines a future noise auto-correlation factor, g_(d)H_(f)H_(f) ^(H), (step 52). A noise auto-correlation device 22 determines a noise auto-correlation factor, E{nn^(H)}, (step 54). A summer 26 sums the two factors together to produce Σ₁, (step 56).

A past input correction device 28 takes the past portion of the channel response matrix, H_(p), and a past determined portion of the data vector, {circumflex over (d)}_(p), to produce a past correction factor, H_(p){circumflex over (d)}_(p), (step 58). A subtractor 30 subtracts the past correction factor from the received vector producing a modified received vector, {tilde over ({circumflex over (r)}, (step 60). An MMSE device 34 uses Σ₁, {tilde over (H)}, and {tilde over ({circumflex over (r)} to determine the received data vector center portion {tilde over ({circumflex over (d)}, such as per Equation 21, (step 62). The next window is determined in the same manner using a portion of {tilde over ({circumflex over (d)} as {circumflex over (d)}_(p) in the next window determination, (step 64). As illustrated in this approach, only data for the portion of interest, {tilde over ({circumflex over (d)}, is determined reducing the complexity involved in the data detection and the truncating of unwanted portions of the data vector.

In another approach to data detection, only the noise term is corrected. In this approach, the system model is per Equation 25.

r={tilde over (H)}{tilde over (d)}+ñ ₂, where ñ₂=H_(p)d_(p)+H_(f)d_(f)+n  Equation 25

Using an MMSE algorithm, the estimated data vector {tilde over ({circumflex over (d)} is per Equation 26.

{tilde over ({circumflex over (d)}=g _(d) {tilde over (H)} ^(H)(g _(d) {tilde over (H)}{tilde over (H)} ^(H)+Σ₂)⁻¹ r  Equation 26

Assuming H_(p)d_(p), H_(f)d_(f) and n are uncorrelated, Equation 27 results.

Σ₂ =g _(d) H _(p) H _(p) ^(H) +g _(d) H _(f) H _(f) ^(H) +E{nn ^(H)}  Equation 27

To reduce the complexity in solving Equation 26 using Equation 27, a full matrix multiplication for H_(p)H_(p) ^(H) and H_(f)H_(f) ^(H) are not necessary, since only the upper and lower corner of H_(p) and H_(f), respectively, are non-zero, in general.

This approach is also described in conjunction with the flow diagram of FIG. 7 and preferred receiver components of FIG. 8, which can be implemented in a WTRU or base station. The circuit of FIG. 8 can be implemented on a single integrated circuit (IC), such as an application specific integrated circuit (ASIC), on multiple IC's, as discrete components or as a combination of IC('s) and discrete components.

A channel estimation device 36 processes the received vector producing the channel estimate matrix portions, H_(p), {tilde over (H)} and H_(f), (step 70). A noise auto-correlation correction device 38 determines a noise auto-correlation correction factor, g_(d)H_(p)H_(p) ^(H)+g_(d)H_(f)H_(f) ^(H), using the future and past portions of the channel response matrix, (step 72). A noise auto correlation device 40 determines a noise auto-correlation factor, E{nn^(H)}, (step 74). A summer 42 adds the noise auto-correlation correction factor to the noise auto-correlation factor to produce Σ₂ (step 76). An MMSE device 44 uses the center portion or the channel response matrix, {tilde over (H)}, the received vector, r, and Σ₂ to estimate the center portion of the data vector, {tilde over ({circumflex over (d)}, (step 78). One advantage to this approach is that a feedback loop using the detected data is not required. As a result, the different slided window version can be determined in parallel and not sequentially.

Discrete Fourier Transform Based Equalization

The sliding window approach described above requires a matrix inversion, which is a complex process. One embodiment for implementing a sliding window utilizes discrete Fourier transforms (DFTs), as follows. Although the preferred implementation of the DFT based approach is with a MMSE algorithm, it can be applied to other algorithms, such as a zero forcing (ZF) based algorithm.

A matrix A_(cir)εC^(N×N), for some integer N, is a circulant matrix if it has the following form per Equation 28.

$\begin{matrix} {A_{cir} = \begin{bmatrix} a_{1} & a_{N} & a_{N - 1} & \; & a_{2} \\ a_{2} & a_{1} & a_{N} & \ddots & \vdots \\ \vdots & a_{2} & a_{1} & \ddots & a_{N - 1} \\ \vdots & \vdots & a_{2} & \ddots & a_{N} \\ a_{N} & a_{N - 1} & \vdots & \; & a_{1} \end{bmatrix}} & {{Equation}\mspace{20mu} 28} \end{matrix}$

This kind of matrix is expressed using the DFT and the IDFT operators, such as per Equation 29.

A _(cir) =F _(N) ⁻¹Λ(A _(cir)[:,1])F _(N)  Equation 29

where, A_(cir)[:,1]=(a₀, a₁, . . . , a_(N))^(T)εC^(N), i.e. it is the first column of matrix A_(cir) Columns other than the first column can be used if properly permuted. F_(N) is the N-point DFT matrix which is defined as, for any xεC^(N), as per Equation 30.

$\begin{matrix} {{\left( {F_{N}x} \right)_{k} = {\sum\limits_{n = 0}^{N - 1}\; {{x(n)}^{j\frac{2\; \pi \; {kn}}{N}}}}}{{k = 0},\ldots \mspace{14mu},{N - 1}}} & {{Equation}\mspace{20mu} 30} \end{matrix}$

F_(N) ⁻¹ is the N-point inverse DFT matrix which is defined as, for any xεC^(N), as per Equation 31.

$\begin{matrix} {{\left( {F_{N}^{- 1}x} \right)_{k} = {{\frac{1}{N}\left( {F_{N}^{*}x} \right)_{k}} = {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}\; {{x(n)}^{{- j}\frac{2\; \pi \; {kn}}{N}}}}}}}{{k = 0},\ldots \mspace{14mu},{N - 1}}} & {{Equation}\mspace{20mu} 31} \end{matrix}$

Λ_(N)(·) is a diagonal matrix, which is defined as, for any xεC^(N), as per Equation 32.

Λ_(N)(x)=diag(F _(N) x)  Equation 32

The inverse of matrix A_(cir) is expressed, such as per Equation 33.

A _(cir) ⁻¹ =F _(N) ⁻¹Λ_(N) ⁻¹(A _(cir)[:,1])F _(N)  Equation 33

The following is an application of a DFT based approach to the data estimation process using the sliding window based chip level equalizer. The first embodiment uses a single receiving antenna. Subsequent embodiments use multiple receiving antennas.

The receiver system is modeled as per Equation 34.

$\begin{matrix} {{r(t)} = {{{\sum\limits_{k = {- \infty}}^{\infty}\; {{d(k)}{h\left( {t - {kT}_{c}} \right)}}} + {n(t)} - \infty} < t < \infty}} & {{Equation}\mspace{20mu} 34} \end{matrix}$

h(·) is the impulse response of the channel. d(k) is the kth transmitted chip samples that is generated by spreading symbols using a spreading code. r(·) is the received signal. n(·) is the sum of additive noise and interference (intra-cell and inter-cell).

Using chip rate sampling and h(·) having a finite support, which means, in discrete-time domain, there is an integer L such that h(i)=0 for i<0 and i≧L, the sampled received signal can be expressed (T_(c) is dropped for simplicity of the notations), as per Equation 35.

$\begin{matrix} {{{r(j)} = {{\sum\limits_{k = 0}^{L - 1}\; {{h(k)}{d\left( {j - k} \right)}}} + {n(j)}}}{j \in \left\{ {\ldots \mspace{14mu},{- 2},{- 1},0,1,2,\ldots}\mspace{14mu} \right\}}} & {{Equation}\mspace{20mu} 35} \end{matrix}$

Based on M (M>L) received signals r(0), . . . , r(M−1), Equation 36 results.

     r = Hd + n      where $\mspace{79mu} {{r = {\left\lbrack {{r(0)},\ldots \mspace{11mu},{r\left( {M - 1} \right)}} \right\rbrack^{T} \in C^{M}}},\mspace{79mu} {d = {\begin{bmatrix} {{d\left( {{- L} + 1} \right)},{d\left( {{- L} + 2} \right)},\ldots \mspace{14mu},} \\ {{d(0)},{d(1)},\ldots \mspace{14mu},{d\left( {M - 1} \right)}} \end{bmatrix}^{T} \in C^{{M + L} = 1}}}}$      n = [n(0), …  , n(M − 1)]^(T) ∈ C^(M)                                      Equation  36 $H = {\left\lbrack \begin{matrix} {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(1)} & {h(0)} & 0 & \cdots & \cdots \\ 0 & {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(1)} & {h(0)} & 0 & \cdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \cdots & \cdots & 0 & {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(1)} & {h(0)} \end{matrix} \right\rbrack  \in C^{M \times {({M + L - 1})}}}$

As illustrated by Equation 36, the H matrix is Toeplitz. As described subsequently in the application for multiple chip rate sampling and/or multiple receive antennas, the H matrix is block Toeplitz. Using the block Toeplitz property, discrete Fourier transform techniques can be applied. The Toeplitz/block Toeplitz nature is produced as a result of the convolution with one channel or the convolution of the input signal with a finite number of effective parallel channels. The effective parallel channels appear as a result of either oversampling or multiple receive antennas. For one channel, a single row is essentially slide down and to the right producing a Toeplitz matrix.

The statistics of the noise vector are treated as having the autocorrelation property, per Equation 37.

E{nn^(H)}=σ²I  Equation 37

The left hand side of equation (5) can be viewed as a “window” of continuous input signal stream. To estimate the data, an approximated model is used. In this approximated model, the first L−1 and the last L−1 elements of vector d are assumed to be zero before applying the MMSE algorithm and the reset M−L+1 elements of d forms a new vector {tilde over (d)}=[d(0), . . . , d(M−L+1)]^(T). The approximated model can be expressed explicitly as per Equation 38.

$\begin{matrix} {{r = {{\overset{\sim}{H}\overset{\sim}{d}} + n}}{where}{\overset{\sim}{H} = {\begin{bmatrix} {h(0)} & 0 & \cdots & \; \\ {h(1)} & {h(0)} & \ddots & \; \\ \vdots & {h(1)} & \ddots & 0 \\ {h\left( {L - 1} \right)} & \vdots & \ddots & {h(0)} \\ 0 & {h\left( {L - 1} \right)} & \ddots & {h(1)} \\ \vdots & 0 & \ddots & \vdots \\ \; & \vdots & \ddots & {h\left( {L - 1} \right)} \end{bmatrix}.}}} & {{Equation}\mspace{20mu} 38} \end{matrix}$

After the vector {tilde over (d)} is estimated, only the middle part of it is taken for de-spreading. Subsequently, the window of observation (i.e. the received signal) is slid by (M−L+1)/2 elements and the process is repeated. FIG. 9 is a graphical representation of the sliding window process, as described above.

Using MMSE algorithm, the estimated data is expressed per Equation 39.

{tilde over ({circumflex over (d)}=R⁻¹{tilde over (H)}^(H)r  Equation 39

where R={tilde over (H)}^(H){tilde over (H)}+σ²I

In Equation 39, neither the matrix R nor the matrix {tilde over (H)} is circulant to facilitate a DFT implementation. To facilitate a DFT implementation, for each sliding step, the approximated system model per Equation 40 is used.

$\begin{matrix} {\mspace{76mu} {{r = {{\overset{\Cup}{H}\overset{\Cup}{d}} + n}}\mspace{76mu} {where}{\overset{\Cup}{H} = {\left\lbrack \begin{matrix} {h(0)} & 0 & \cdots & \; & \; & \; & \; \\ {h(1)} & {h(0)} & \ddots & \; & \; & \; & \; \\ \vdots & {h(1)} & \ddots & 0 & \; & \; & \; \\ {h\left( {L - 1} \right)} & \vdots & \ddots & {h(0)} & 0 & \; & \; \\ 0 & {h\left( {L - 1} \right)} & \ddots & {h(1)} & {h(0)} & \ddots & \; \\ \vdots & 0 & \ddots & \vdots & \vdots & \ddots & 0 \\ \; & \vdots & \ddots & {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(0)} \end{matrix} \right\rbrack  \in C^{M \times M}}}\mspace{79mu} {\overset{\Cup}{d} = {\left\lbrack {{d(0)},\ldots \mspace{14mu},{d\left( {M - 1} \right)}} \right\rbrack^{T} \in C^{M \times 1}}}}} & {{Equation}\mspace{20mu} 40} \end{matrix}$

In Equation 40, only the first L−1 elements (equations) are approximations of those of Equation 36.

The matrix {hacek over (H)} is replaces by a circulant matrix, such as per Equation 41.

$\begin{matrix} \; & {\mspace{625mu} {{Equation}\mspace{14mu} 41}} \end{matrix}$ $\mspace{79mu} {H_{cir} = \begin{bmatrix} {h(0)} & 0 & \cdots & 0 & {h\left( {L - 1} \right)} & \cdots & {h(1)} \\ {h(1)} & {h(0)} & \ddots & \vdots & 0 & \ddots & \vdots \\ \vdots & {h(1)} & \ddots & 0 & \vdots & \; & {h\left( {L - 1} \right)} \\ {h\left( {L - 1} \right)} & \vdots & \ddots & {h(0)} & 0 & \ddots & 0 \\ 0 & {h\left( {L - 1} \right)} & \ddots & {h(1)} & {h(0)} & \ddots & \vdots \\ \vdots & 0 & \ddots & \vdots & \vdots & \ddots & 0 \\ 0 & \vdots & \ddots & {h\left( {L - 1} \right)} & {h\left( {L - 2} \right)} & \cdots & {h(0)} \end{bmatrix}}$

The system model, for each sliding step, is per Equation 42.

r=H _(cir) d+n  Equation 42

with d=[d(0), . . . , d(M−1)]^(T)εC^(M×1) The vector d in Equation 42, due to the new model, is different than the vector d in Equation 36. Equation 42 adds additional distortion to the first L−1 element of Equation 39. This distortion makes the two ends of the estimated vector d inaccurate. FIG. 10 is a graphical representation of the model construction process.

Using approximated model per Equation 42, the MMSE algorithm yields the estimated data as per Equation 43.

{circumflex over (d)}=R_(cir) ⁻¹H_(cir) ^(H)r  Equation 43

where R_(cir)=H_(cir) ^(H)H_(cir)+σ²I Both H_(cir) ^(H) and R_(cir) are circulant and R_(cir) is of the form per Equation 44.

$\begin{matrix} {R_{cir} =} & {\mspace{565mu} {{Equation}\mspace{14mu} 44}} \end{matrix}\left\lbrack \begin{matrix} R_{0} & R_{1} & \cdots & R_{L - 1} & 0 & 0 & \cdots & \; & \; & \; & R_{2}^{*} & R_{1}^{*} \\ R_{1}^{*} & R_{0} & \ddots & \; & R_{L - 1} & 0 & \ddots & \; & \; & \; & \vdots & \; \\ \vdots & R_{1}^{*} & \ddots & R_{1} & \; & R_{L - 1} & \ddots & 0 & \; & \; & R_{L - 1}^{*} & \vdots \\ R_{L - 1}^{*} & \; & \ddots & R_{0} & R_{1} & \; & \ddots & 0 & 0 & \; & 0 & R_{L - 1}^{*} \\ 0 & R_{L - 1}^{*} & \; & R_{1}^{*} & R_{0} & \ddots & \; & R_{L - 1} & 0 & \; & \vdots & 0 \\ \vdots & 0 & \ddots & \; & R_{1}^{*} & \ddots & \ddots & \; & R_{L - 1} & \ddots & 0 & \vdots \\ 0 & \vdots & \ddots & R_{L - 1}^{*} & \; & \ddots & \ddots & R_{1} & \; & \ddots & 0 & 0 \\ R_{L - 1} & 0 & \ddots & 0 & R_{L - 1}^{*} & \; & \ddots & R_{0} & R_{1} & \; & R_{L - 1} & 0 \\ \vdots & R_{L - 1} & \; & \vdots & 0 & R_{L - 1}^{*} & \; & R_{1}^{*} & R_{0} & \ddots & \; & R_{L - 1} \\ \vdots & \vdots & \; & \; & 0 & 0 & \ddots & \; & R_{1}^{*} & \ddots & R_{1} & \; \\ \vdots & \vdots & \ddots & \; & \; & 0 & \ddots & R_{L - 1}^{*} & \; & \ddots & R_{0} & R_{1} \\ R_{1} & R_{2} & \cdots & 0 & \; & \cdots & 0 & 0 & R_{L - 1}^{*} & \cdots & R_{1}^{*} & R_{0} \end{matrix} \right\rbrack$

Applying the properties of circulant matrices, the estimated data is per Equation 45.

{circumflex over (d)}=F _(M) ⁻¹Λ_(M) ⁻¹(R _(cir)[:,1])Λ_(M)(H _(cir) ^(H)[:,1])F _(M) r  Equation 45

FIG. 11 is a diagram of a circuit for estimating the data per Equation 45. The circuit of FIG. 11 can be implemented on a single integrated circuit (IC), such as an application specific integrated circuit (ASIC), on multiple IC's, as discrete components or as a combination of IC('s) and discrete components.

The estimated channel response {tilde over (H)} is processed by an {hacek over (H)} determination device 80 to determine the Toeplitz matrix {hacek over (H)}. A circulant approximation device 82 processes {hacek over (H)} to produce a circulant matrix H_(cir). A Hermitian device 84 produces the Hermitian of H_(cir), H_(cir) ^(H). Using H_(cir), H_(cir) ^(H) and the noise variance σ², R_(cir) is determined by a R_(cir) determining device 86. Using a first column of H_(cir) ^(H), a diagonal matrix is determined by a Λ_(M)(H_(cir) ^(H)[:,1]) determining device 88. Using a first column of R_(cir), an inverse diagonal matrix is determined by a Λ_(M) ⁻¹(R_(cir)[:,1]) determination device 90. A discrete Fourier transform device 92 performs a transform on the received vector, r. The diagonal, inverse diagonal and Fourier transform result are multiplied together by a multiplier 96. An inverse Fourier transform device 94 takes an inverse transform of the result of the multiplication to produce the data vector {tilde over (d)}.

The sliding window approach is based on an assumption that the channel is invariant within each sliding window. The channel impulse response near the beginning of the sliding window may be used for each sliding step.

One preferred approach for determining the window step size N_(ss) and window size M is per Equation 46, although others may be used.

N _(ss)=2N _(symbol)×SF and M=4N _(symbol)×SF  Equation 46

N_(symbol)ε{1, 2, . . . } is the number of symbols and is a design parameter which should be selected, such that M>L. Since M is also the parameter for DFT which may be implemented using FFT algorithm. M may be made large enough such that the radix-2 FFT or a prime factor algorithm (PFA) FFT can be applied. After the data is estimated, 2N_(symbol)×SF samples are taken to process despreading starting from N_(symbol)×SF^(th) sample. FIG. 11 is an illustration of taking the samples for despreading.

Multiple Receive Antenna Equalization

The following is an embodiment using multiple receive antennas, such as K receive antennas. Samples of the received vector and estimates of the channel impulse response are taken for each antenna independently. Following the same process as for the single antenna embodiment, each antenna input, r_(k), is approximated per Equation 47.

r _(k) =H _(cir,k) d+n _(k) for k=1, . . . , K  Equation 47

or in block matrix form per Equation 48

$\begin{matrix} {\begin{bmatrix} r_{1} \\ r_{2} \\ \vdots \\ r_{K} \end{bmatrix} = {{\begin{bmatrix} H_{{cir},1} \\ H_{{cir},2} \\ \vdots \\ H_{{cir},K} \end{bmatrix}d} + \begin{bmatrix} n_{1} \\ n_{2} \\ \vdots \\ n_{K} \end{bmatrix}}} & {{Equation}\mspace{20mu} 48} \end{matrix}$

Equations 49 and 50 are estimates of the auto-correlation and cross-correlation properties of the noise terms.

E{n_(k)n_(k) ^(H)}=σ²I for k=1, . . . , K  Equation 49

and

E{n_(k)n_(j) ^(H)}=0 for k≠j  Equation 50

Applying MMSE algorithm, the estimated data can be expressed as per Equation 51.

$\begin{matrix} {{\hat{d} = {R_{cir}^{- 1}{\sum\limits_{k = 1}^{K}\; {H_{{cir},k}^{H}r_{k}}}}}{where}{R_{cir} = {{\sum\limits_{k = 1}^{K}\; {H_{{cir},k}^{H}H_{{cir},k}}} + {\sigma^{2}I}}}} & {{Equation}\mspace{20mu} 51} \end{matrix}$

R_(cir) is still a circulant matrix and the estimated data can be determined per Equation 52.

$\begin{matrix} {\hat{d} = {F_{M}^{- 1}{\Lambda_{M}^{- 1}\left( {R_{cir}\left\lbrack {:{,1}} \right\rbrack} \right)}{\sum\limits_{k = 1}^{K}\; {{\Lambda_{M}\left( {H_{{cir},k}^{H}\left\lbrack {:{,1}} \right\rbrack} \right)}F_{M}r_{k}}}}} & {{Equation}\mspace{20mu} 52} \end{matrix}$

If the receive antennas are positioned close to each other, the noise terms may be correlated in both time and space. As a result, some degradation in the performance may result.

Multiple Chip Rate Sampling (Oversampling) Equalization

The following describes embodiments using a sliding window based equalization approach with multiple chip rate sampling. Multiple chip rate sampling is when the channel is sampled at a sampling rate which is an integer multiple of the chip rate, such as two times, three times, etc. Although the following concentrates on two times per chip sampling, these approaches can be applied to other multiples.

Using a sliding window of width of N chips and two time chip rate sampling, our received vector is r=[r₀, r₁, . . . , r_(2N−1)]^(T). This vector may be rearranged and separated into an even received vector r_(e)=[r₀, r₂, . . . , r_(2N−2)]^(T) and an odd received vector r_(o)=[r₁, r₃, . . . , r_(2N−1)]^(T), with r=[r_(e), r_(o)]^(T). Without loss of generality, the data transmission model is per Equation 53.

$\begin{matrix} {\begin{bmatrix} r_{e} \\ r_{o} \end{bmatrix} = {{\begin{bmatrix} H_{e} \\ H_{o} \end{bmatrix}d} + \begin{bmatrix} n_{e} \\ n_{o} \end{bmatrix}}} & {{Equation}\mspace{20mu} 53} \end{matrix}$

Equation 53 separates the effective 2-sample-per-chip discrete-time channel into two chip-rate discrete-time channels.

The matrices H_(e) and H_(o) in Equation 53 are, correspondingly, the even and odd channel response matrices. These matrices are constructed from the even and odd channel response vectors h_(e) and h_(o), which are obtained by sampling the channel response at 2 samples per chip and separating it into the even and odd channel response vectors.

The channel noise is modeled as white with a variance σ², as per Equation 54.

E[n_(e)n_(e) ^(H)]=E[n_(o)n_(o) ^(H)]=σ²I  Equation 54

If the channel is an additive white Gaussian noise (AWGN) channel and the received data is provided directly from the sampled channel, then Equation 55 results.

E[n_(e)n_(o) ^(H)]=0  Equation 55

As a result, the problem is mathematically similar to the case of the chip-rate equalizer for 2 receive antennas with uncorrelated noise, as previously described. However, the received antenna signals in many implementations are processed by a receive-side root-raised cosine (RRC) filter before being provided to the digital receiver logic for further processing. Following such processing, the received noise vector is no longer white, but has a raised-cosine (RC) autocorrelation function. RC is the frequency-domain square of a RRC response. Since the RC pulse is a Nyquist pulse, Equation 54 holds, however Equation 55 does not. The (i,j)^(th) element of the matrix

$\Lambda_{cross}\overset{def}{=}{\frac{1}{\sigma^{2}}{E\left\lbrack {n_{e}n_{o}^{H}} \right\rbrack}}$

is per Equation 56.

$\begin{matrix} {{\frac{1}{\sigma^{2}}{E\left\lbrack {n_{e}n_{o}^{H}} \right\rbrack}_{({i,j})}} = {x_{RC}\left( {{{i - j}} + 0.5} \right)}} & {{Equation}\mspace{20mu} 56} \end{matrix}$

x_(RC) is the unity-symbol-time normalized RC pulse shape.

Properties of Λ_(cross) are it is real, symmetric and Toeplitz; it is not banded and has no zero entries and its entries do get smaller and tend to 0 as they get farther and farther away from the main diagonal.

Σ_(n) represent the cross-correlation matrix of the total noise vector and is per Equation 57.

$\begin{matrix} {\Sigma_{n} = {\sigma^{2}\begin{bmatrix} I & \Sigma_{cross} \\ \Sigma_{cross} & I \end{bmatrix}}} & {{Equation}\mspace{20mu} 57} \end{matrix}$

Exact Solution

The exact solution to the problem of linear minimum mean-square estimation of d from the observation of r is per Equation 58.

{circumflex over (d)} _(MMSE)=(H ^(H)Σ_(n) ⁻¹ H+I)⁻¹ H ^(H)Σ_(n) ⁻¹ r  Equation 58

where y=H^(H)Σ_(n) ⁻¹r is the whitening matched filtering (WMF)

{circumflex over (d)}_(MMSE)=(H^(H)Σ_(n) ⁻¹H+I)⁻¹y is the linear MMSE equalization

Neither H^(H)Σ_(n) ⁻¹ nor H^(H)Σ_(n) ⁻¹H+I are Toeplitz and neither can be made Toeplitz through elemental unitary operations (e.g. row/column re-arrangements), due to the structure of Σ_(n). Accordingly, DFT-based methods based on circulant approximations of Toeplitz matrices cannot be applied here and an exact solution is highly complex.

Two embodiments for deriving an efficient algorithm for solving this problem are described. The first embodiment uses a simple approximation and the second embodiment uses an almost-exact solution.

Simple Approximation

The simple approximation ignores the correlation between n_(e) and n_(o), Σ_(cross)=0. As a result, the same approach as multiple chip-rate receive antennas is used.

The complexity of this simple approximation approach is as follows. N-chip data blocks are considered. For rough approximation, an N-point DFT complexity, given by N log N operations per second (ops), is assumed. Additionally, N-point vector multiplications are assumed to take N ops and vector additions are ignored.

The complexity of the DFT-based approach can be roughly partitioned into 2 components: the processing which has to be performed on every received data set and the processing which is performed when the channel estimate is updated, which is typically done one to two orders of magnitude less frequently then the former operation.

For processing performed on each received data set, the following operations are performed: 2 N-point DFTs to transform the received vector into the frequency domain; 2 N-point vector multiplications (multiply each received vector by the appropriate “state” vector”); and one more DFT to transform the result back into time domain. Thus, the approximate complexity is per Equation 59.

C _(1,r)=3N log N+2N  Equation 59

For processing performed when the channel response is updated, the following operations are performed: 2 DFT operations, 6 N-point vector multiplies and a vector division, which need to be taken 10 times the operations of a vector multiply. Thus, the complexity of this step is roughly given per Equation 60.

C _(1,r)=2N log N+16N  Equation 60

Almost Exact Solution

For the almost-exact solution which uses a block-Toeplitz solution, the vector and matrices are rearranged in their natural order, i.e., the natural order being an order that elements of the received vector and channel response matrix were actually received, such that the vector r is given by r=[r₀, r₁, . . . , r_(2N−1)]^(T). Equation 61 is the natural order model.

r=H _(bT) d+n

where H_(bT) is defined as

$\begin{matrix} {H_{bT} = {\begin{bmatrix} h_{e,1} \\ h_{o,1} \\ \vdots \\ h_{o,N} \end{bmatrix} = \begin{bmatrix} G_{1} \\ G_{2} \\ \vdots \\ G_{N} \end{bmatrix}}} & {{Equation}\mspace{20mu} 61} \end{matrix}$

h_(e,i) is the i^(th) row of H_(e) and h_(o,i) is the i^(th) row of H_(o). G_(i) is a 2×N matrix whose 1^(st) row is h_(e,i) and whose 2^(nd) row is h_(o,i). Using G_(i) [x,y] as the row-x, column-y element of H_(bT) is block-Toeplitz as illustrated in Equation 62.

G _(i) [x,y]=G _(j) [x,y+(i−j)]  Equation 62

provided that 1≦y+(i−j)≦N

The block-Toeplitz structure of H_(bT) follows immediately from the Toeplitz structure of H_(e) and H_(o) and the row-rearrangement. From the Toeplitz structure of I and Σ_(cross), the autocorrelation matrix of the noise in the re-defined problem is also block Toeplitz. Because this matrix is also symmetric, it can be rewritten per Equation 63.

Σ_(bT)=[Σ_(i,j)]_(1≦i,j≦N)  Equation 63

where Σ_(i,j) are 2×2 matrices with the property that Σ_(i,j)=Σ_(|i−j|)

Subsequently, block-circulant approximations to the block-Toeplitz matrices are produced. Since the H_(bT) matrix is also banded, the block circulant approximation of H_(bT) is then obtained directly. However, Σ_(bT) is not banded and therefore it is not possible to produce a block-circulant approximation directly from it. Since the elements of Λ_(bT) tend to 0 as they get farther away from the main diagonal, a banded approximation to Σ_(bT) is per Equation 64.

Σ_(bT)≈{tilde over (Σ)}_(bT)=[{tilde over (Σ)}_(i,j)]_(1≦i,j≦N)  Equation 64

where {tilde over (Σ)}_(i,j) are 2×2 matrices with the property that

{tilde over (Σ)}_(i,j)=Σ_(|i−j|) if |i−j|≦B_(n) and {tilde over (Σ)}_(i,j)=0 otherwise

The noise-covariance-bandwidth, B_(n), is a design parameters that is selected. Due to the decay properties of the RC pulse shape, it is likely to be only several chip. Now {tilde over (Σ)}_(bT) is banded block-Toeplitz and a circulant approximation to it is produced.

The circulant approximations of H_(bT) and {tilde over (Σ)}_(bT) are H_(bC) and Σ_(bC), respectively. W_(n) denotes the n-point DFT matrix, that is if x is an n-vector, then x_(f)=W_(n)x is the DFT of x. A block-circulant matrix C is of the form of Equation 65.

$C = \begin{bmatrix} C_{1} & C_{2} & \cdots & C_{M} \\ C_{2} & C_{3} & \cdots & C_{1} \\ \vdots & \vdots & ⋰ & \vdots \\ C_{M} & C_{1} & \cdots & C_{M - 1} \end{bmatrix}$

where C_(i) is an N×N matrix and therefore C is an MN×MN matrix

C can also be written as Equation 66.

C=W _(M×N) ⁻¹Λ_(M×N)(C)W _(M×N)  Equation 66

where W_(M×N) is the block-N-DFT matrix defined as W_(M×N)=W_(M){circle around (x)}I_(N) Λ_(M×N)(C) is a block diagonal matrix that depends on C and is given by Equation 67.

$\begin{matrix} {{\Lambda_{M \times N}(C)} = \begin{bmatrix} {\Lambda_{1}(C)} & \; & \; & \; \\ \; & {\Lambda_{2}(C)} & \; & \; \\ \; & \; & ⋰ & \; \\ \; & \; & \; & {\Lambda_{M}(C)} \end{bmatrix}} & {{Equation}\mspace{20mu} 67} \end{matrix}$

Λ_(i)(C) is an N×N matrix. To completely specify Λ_(i)(C), λ_(i,(k,l)) denotes the (k,l)^(th) element of Λ_(i)(C) and is defined as

$\lambda_{({k,l})}\overset{def}{=}{\left\lbrack {\lambda_{1,{({k,l})}},\lambda_{2,{({k,l})}},\ldots \mspace{14mu},\lambda_{M,{({k,l})}}} \right\rbrack^{T} \cdot c_{i,{({k,l})}}}$

denotes the (k,l)^(th) element of C and is defined as

$c_{({k,l})}\overset{def}{=}{\left\lbrack {c_{1,{({k,l})}},c_{2,{({k,l})}},\ldots \mspace{14mu},c_{M,{({k,l})}}} \right\rbrack^{T} \cdot \lambda_{({k,l})}}$

is the M-point DFT of c_((k,l)) and is per Equation 68.

λ_((k,l))=W_(M) ^(c) _((k,l))  Equation 68

Equations 66-68 specify the block-DFT representation of square block circulant matrices. N² DFTs are required to compute Λ_(M×N)(C).

The MMSE estimator is rewritten per Equation 69.

{circumflex over (d)} _(MMSE) =H ^(H)(Σ_(n) +HH ^(H))⁻¹ r  Equation 69

The MMSE estimator form as per Equation 68 has several advantages. It requires only a single inverse matrix computation and thus in the DFT domain only a single vector division. This provides a potentially significant savings as divisions are highly complex.

The almost-exact solution has two steps in the most preferred embodiment, although other approaches may be used. Every time a new channel estimate is obtained, the channel filter is updated, (H^(H)(Σ_(n)+HH^(H))⁻¹ is determined). For every data block, this filter is applied to the received data block. This partition is utilized because the channel is updated very infrequently compared to the received data block processing and therefore significant complexity reduction can achieved by separating the overall process into these two steps.

The DFT of Σ_(n) is the DFT of the pulse shaping filter multiplied by the noise variance σ². Since the pulse shaping filter is typically a fixed feature of the system, its DFT can be precomputed and stored in memory and thus only the value σ² is updated. Since the pulse-shaping filter is likely to be close to the “ideal” (IIR) pulse shape, the DFT of the ideal pulse shape can be used for Σ_(n), reducing the complexity and is also far away from the carrier.

To channel update step, the following is performed:

-   -   1. The “block-DFT” of H needs to be computed. Since the block is         of width 2, it requires 2 DFTs. The result is a N×2 matrix whose         rows are the DFTs of h_(e) and h_(o).     -   2. The “block-DFT” of HH^(H) is computed by finding         element-by-element autocorrelations and the crosscorrelation of         h_(e) and h_(o). This required 6N complex multiplies and 2N         complex adds: the products of N 2×2 matrices are computed with         there own Hermitian transposes.     -   3. The block-DFT of Σ_(n) is added, which requires 3N multiplies         (scale the stored block-DFT of the RRC filter by σ²) and 3N adds         to add the block-DFT of the two matrices.     -   4. An inverse of Σ_(n)+HH^(H) is taken in the block-DFT domain.         To do this an inverse of each of the N 2×2 matrices is taken in         the block-DFT domain. To estimate the total number of         operations, consider a Hermitian matrix

$M = {\begin{bmatrix} a & b \\ b^{*} & a \end{bmatrix}.}$

The inverse of this matrix is given per Equation 70.

$\begin{matrix} {M^{- 1} = {\frac{1}{a^{2} - {b}^{2}}\begin{bmatrix} a & {- b} \\ {- b^{*}} & a \end{bmatrix}}} & {{Equation}\mspace{20mu} 70} \end{matrix}$

-   -   Accordingly, the complexity of computing each inverse involves 3         real multiplications and 1 real subtraction (roughly 1 complex         multiply) and 1 real division.     -   5. The result are block-multiplied by the block-DFT of H^(H),         which, takes a total of 8N multiplies+4N adds (since H^(H) is         not Hermitian).

To summarize, the following computation are required: 2 N-point DFTs; 18N complex multiplies (17 N-point vector multiplies+N stand-alone multiplies); 11N complex adds (11 N-point vector adds); and 1N real divisions.

The complexity of processing a data block r of 2N values (N chips long) involves: 2 N-point DFTs; one product of the N-point block-DFTs (filter and data), which required 8N complex multiplies and 4N complex adds; and 1 N-point inverse DFTs.

To summarize, the following is required: 3 N-point DFTs; 8N complex multiplies (8 N-point vector multiplies); and 4N complex adds (4 N-point vector adds).

Multiple Chip Rate Sampling and Multiple Receive Antenna Equalization

The following are embodiments using multiple chip rate sampling and multiple receive antennas. For L receive antennas, 2L channel matrices—one “even” and one “odd” matrix for each antenna result. The channel matrices for l^(th) antenna are denoted as H_(l,e) and H_(l,o) and h_(l,e,n) and h_(l,o,n) denote the n^(th) row of such matrix. Each channel matrix is Toeplitz and with the appropriate re-arrangement of rows the joint channel matrix is a block-Toeplitz matrix, per Equation 71.

$\begin{matrix} {H_{bT} = {\begin{bmatrix} h_{1,e,1} \\ h_{1,o,1} \\ \vdots \\ h_{L,o,N} \end{bmatrix} = \begin{bmatrix} G_{1} \\ G_{2} \\ \vdots \\ G_{N} \end{bmatrix}}} & {{Equation}\mspace{20mu} 71} \end{matrix}$

The matrices G_(i) are the Toeplitz blocks of H_(bT). Each G_(i) is a 2L×N matrix.

Estimating the vector d from the received observations r can be modeled per Equation 72.

r=H _(bT) d+n  Equation 72

The MMSE estimation formulation is per Equation 73.

{circumflex over (d)} _(MMSE) =H _(bT) ^(H)(Σ_(n) +H _(bT) H _(bT) ^(H))⁻¹ r  Equation 73

Σ_(n) is the covariance of the noise vector n. The form that the solution of Equation 73 depends on the assumptions that are made for Σ_(n). The introduction of the multiple receive antenna introduces an additional spatial dimension. Although the interplay of temporal and spatial correlations can be extremely complex, it can be assumed that the spatial correlation properties of the noise do not interplay with the temporal correlation properties, except as a direct product of the two, as per Equation 74.

Σ_(n)=Σ_(n,1ant)

Σ_(sp)  Equation 74

E_(n,1ant) is the noise covariance matrix of the noise observed at a single antenna as per Equation 57. Σ_(n,1ant) is of dimension 2N×2N. Σ_(sp) is the normalized synchronous spatial covariance matrix, i.e. it is the covariance matrix between the L noise samples observed at the L antennas at the same time normalized to have 1's on the main diagonal.

denotes the Kroenecker product.

Σ_(n) is a 2LN×2LN Hermitian positive semi-definite matrix, which is block-Toeplitz with 2L×2L blocks. To estimate the data, four preferred embodiments are described: an exact solution; a simplification by assuming that the L receive antenna have uncorrelated noise; a simplification by ignoring the temporal correlation of the noise in the odd and even streams from the same antenna; and a simplification by assuming that all 2L chip-rate noise streams are uncorrelated.

The complexity of DFT-based processing using the circulant approximation may be partitioned into two components: the processing of channel estimation which need not be done for every new data block and the processing of data itself which is performed for every data block. In all four embodiments, the complexity of processing data involves: 2L forward N-point DFTs; 2LN complex multiplies; and 1 inverse N-point DFT. The complexity of processing the channel estimate varies for each embodiment.

In the case of the exact MMSE solution, the complexity of computing the “MMSE filter” from the channel estimate is as follows: 2L N-point DFT's; N 2L×2L matrix products+N 2L×2L matrix additions to compute (Σ_(n)+H_(bT)H_(bT) ^(H)); N 2L×2L matrix inverses to compute the inverse of (Σ_(n)+H_(bT)H_(bT) ^(H)); and N 2L×2L matrix products to produce the actual filter.

A major contributor to the overall complexity of this process is the matrix inverse step in which an inverse of 2L×2L matrices has to be taken. It is precisely this complexity that can be reduced by various assumptions on the uncorrelated nature of the noise, as follows:

-   -   1. If it is assumed that the noise is uncorrelated both         temporally (odd/even samples) and spatially (across antennas),         then Σ_(n) reduces to a diagonal matrix and the problem is         identical to single-sample-per-chip sampling with 2L antennas         with spatially uncorrelated noise. As a result, the operation of         matrix inverse simply reduces to a division since all the         matrices involved are Toeplitz.     -   2. If it is assumed that the noise is spatially uncorrelated,         then the matrix inverses involved are those of 2×2 matrices.     -   3. If it is assumed that a temporal uncorrelation of odd/even         streams but a spatial noise correlation is retained, the matrix         inverses involved are L×L. 

1. An apparatus for use in wireless communication comprising: a receiver configured to transform a received wireless communications signal to produce a received vector by sampling at a multiple of a data signal chip rate; and a processor configured to process the received vector using a sliding window based approach, such that for each processing window in a plurality of processing windows an approximate circulant channel response matrix is produced and used to estimate a data vector corresponding to the window.
 2. (canceled)
 3. The apparatus of claim 1, further comprising: a root-raised cosine filtering unit configured to apply a root-raised cosine filter to the received vector.
 4. The apparatus of claim 1, wherein the processor is configured to ignore noise cross correlation.
 5. The apparatus of claim 1, wherein the processor is configured to use the received vector and the approximate circulant channel response matrix arranged in a natural order.
 6. The apparatus of claim 1, wherein the receiver is configured to transform a plurality of received wireless communications signals from a plurality of antennas to produce a received vector.
 7. The apparatus of claim 1, wherein the processor is configured to multiply a discrete Fourier transform of a pulse shaping filter by a measured noise variance to produce a discrete Fourier transform of the noise vector cross correlation.
 8. The apparatus of claim 1, wherein the processor is configured to multiply a discrete Fourier transform of an ideal pulse shape by a measured noise variance to produce a discrete Fourier transform of the noise vector cross correlation.
 9. The apparatus of claim 1, further comprising: a summer, configured to combine the data vector corresponding to each window to form a combined data vector.
 10. The apparatus of claim 1 configured as a wireless transmit/receive unit (WTRU).
 11. The apparatus of claim 1 configured as a base station.
 12. A method for use in wireless communications, the method comprising: transforming a received wireless communications signal to produce a received vector by sampling at a multiple of a data signal chip rate; and processing the received vector using a sliding window based approach, such that for each processing window in a plurality of processing windows an approximate circulant channel response matrix is produced and used to estimate a data vector corresponding to the window.
 13. (canceled)
 14. The method of claim 12, further comprising: applying a root-raised cosine filter to the received vector.
 15. The method of claim 12, wherein the processing includes ignoring noise cross correlation.
 16. The method of claim 12, wherein the processing includes using the received vector and the approximate circulant channel response matrix arranged in a natural order.
 17. The method of claim 12, wherein the transforming includes producing a plurality of received vectors corresponding to a plurality of received wireless communications signals from a plurality of antennas.
 18. The method of claim 12, wherein the processing includes multiplying a discrete Fourier transform of the pulse shaping filter by a measured noise variance to produce a discrete Fourier transform of the noise vector cross correlation.
 19. The method of claim 12, wherein the processing includes multiplying a discrete Fourier transform of an ideal pulse shape by a measured noise variance to produce a discrete Fourier transform of the noise vector cross correlation.
 20. The method of claim 12, further comprising: combining the data vector corresponding to each window to form a combined data vector. 